\newproblem{lay:4_4_19}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 4.4.19}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Let $S$ be a finite set of in a vector space $V$ with the property that every $\mathbf{x}$ in $V$ has a unique representation as a linear
	combination of the elements of $S$. Show that $S$ is a basis of $V$.
}{
  % Solution
	The fact that every $\mathbf{x}$ in $V$ has a representation as a linear combination of elements of $S$ means that $S$ spans $V$. The fact that
	this representation is unique implies that the set $S$ is linearly independent. These are the two conditions to become a basis and, therefore, the set
	$S$ is a basis of $V$.
}
\useproblem{lay:4_4_19}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
